Convex: Theory, Methods & Applications

• Convex is a property of sets and functions ensuring every line segment between two points lies entirely within the set, underpinning diverse mathematical and algebraic frameworks.
• Advanced formulations of convexity encompass intrinsic mixing operations, categorical models via Giry monads, and convex envelopes that facilitate rigorous analysis and tractable optimization.
• Applications of convexity span optimization algorithms, graph theory, discrete tomography, and imaging, enabling robust, efficient problem-solving across disciplines.

Convex denotes a family of closure phenomena that preserve admissible combinations. In Euclidean and affine settings, a set is convex when it contains every segment between its points; in abstract algebraic settings, convexity is the existence of coherent mixing operations; in combinatorics it appears through closure systems with anti-exchange; in graph theory and network science it is expressed through geodesic containment; and in optimization it governs convex functions, convex envelopes, convex relaxations, and tractable modeling languages (0903.5522, Marc et al., 2016, Udell et al., 2014).

1. Abstract and algebraic formulations

A central generalization replaces ambient vector-space structure by intrinsic mixing. In Fritz’s formulation, a convex space is a set $C$ equipped with binary operations

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$

satisfying the unit law $cc_0(x,y)=y$, idempotency $cc_\lambda(x,x)=x$, parametric commutativity $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$, and deformed parametric associativity

$$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$

when $\lambda\mu\neq 1$ (0903.5522). Writing $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$, the structure isolates coherent finite mixing rather than linear addition itself.

The same notion has two equivalent categorical descriptions. First, convex spaces are algebras over the finitary Giry monad: for a set $X$,

$$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$

and a convex structure on $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$0 is an evaluation map $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$1 satisfying the unit and associativity law

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$2

Second, convex spaces are precisely models of the Lawvere theory of finite stochastic maps $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$3; Proposition 3.7 identifies the binary-operation, monad-algebra, and Lawvere-theoretic viewpoints as equivalent (0903.5522).

This abstract definition recovers ordinary convex subsets of real vector spaces and also includes non-geometric examples. If $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$4 is a convex subset of a real vector space, then $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$5 yields a convex space. But convex spaces need not embed into vector spaces: semilattices form convex spaces of combinatorial type, where every nontrivial convex combination of two points is constant in $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$6, and the resulting operation is exactly an idempotent, commutative, associative meet (0903.5522). The same paper interprets convex subsets of vector spaces as probabilistic and semilattices as possibilistic, with convex spaces unifying both.

A formalized presentation in Coq develops the same idea as intrinsic barycentric structure. There a convex space carries operations $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$7 for $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$8, and every such space embeds into a conical space

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$9

with embedding $cc_0(x,y)=y$0 satisfying

$cc_0(x,y)=y$1

This embedding linearizes barycentric identities and supports formal development of convex hulls, convex subsets, and convex functions on distribution spaces (Affeldt et al., 2020).

2. Convex functions, envelopes, and algebraic certificates

On ordered convex spaces, a function $cc_0(x,y)=y$2 is convex when

$cc_0(x,y)=y$3

for all $cc_0(x,y)=y$4 and $cc_0(x,y)=y$5; concavity is obtained by reversing the order (Affeldt et al., 2020). In polynomial optimization, convexity is expressed through the Hessian: a polynomial $cc_0(x,y)=y$6 is convex iff $cc_0(x,y)=y$7 for all $cc_0(x,y)=y$8, equivalently $cc_0(x,y)=y$9 for all $cc_\lambda(x,x)=x$0 (Ahmadi et al., 2024).

A central convex-analytic construction is the convex envelope

$cc_\lambda(x,x)=x$1

For a continuously differentiable ray-concave function $cc_\lambda(x,x)=x$2 on a polytope $cc_\lambda(x,x)=x$3, convex on every facet, the paper on ray-concave functions defines

$cc_\lambda(x,x)=x$4

where $cc_\lambda(x,x)=x$5 are the two boundary intersections of the ray through $cc_\lambda(x,x)=x$6, and proves that if $cc_\lambda(x,x)=x$7 is positively homogeneous then $cc_\lambda(x,x)=x$8 (Barrera et al., 2021). This yields explicit envelopes over arbitrary polytopes and includes a previously unknown envelope for the probability/reliability function $cc_\lambda(x,x)=x$9.

A complementary algebraic certificate is sos-convexity. A polynomial is sos-convex when $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$0 is a sum of squares in $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$1, equivalently $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$2 for some polynomial matrix $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$3. For ternary quartic forms, the paper “Convex Ternary Quartics Are SOS-Convex” proves the exact equality

$cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$4

that is, every convex quartic form in three variables is sos-convex (Ahmadi et al., 2024). The result is presented as a convex analogue of Hilbert’s theorem for nonnegative ternary quartics, and the paper shows that exploiting the special linear compatibility relations of Hessian biquadratic forms is essential.

3. Optimization, projection, and computational frameworks

Convexity also governs the shape of optimization trajectories. For gradient descent on a convex $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$5-smooth function,

$cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$6

the optimization curve $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$7 is provably convex for

$cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$8

while monotone decrease of function values still holds on the larger interval $cc_\lambda(x,y)=cc_{1-\lambda}(y,x)$9 (Barzilai et al., 13 Mar 2025). The same paper constructs a one-dimensional convex $$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$0-smooth counterexample showing that for every $$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$1 and every

$$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$2

the optimization curve can be non-convex even though the objective still decreases monotonically. In contrast, for gradient flow $$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$3, the objective curve $$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$4 is convex for every convex $$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$5-smooth $$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$6, and the gradient norm is nonincreasing in both discrete and continuous time on the full natural stability range (Barzilai et al., 13 Mar 2025).

At the modeling-language level, Convex.jl treats convex programs as abstract syntax trees whose nodes carry sign, curvature, monotonicity, evaluability, and conic-form metadata. It checks disciplined convex programming compliance, canonicalizes to conic form, and dispatches to LP, SOCP, SDP, or exponential-cone solvers through Julia’s multiple dispatch (Udell et al., 2014). The target conic form is

$$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$7

and each atom is equipped with a graph-form template. The framework is explicitly presented as a convex optimization modeling system whose separation of atoms from methods makes extension by new convex primitives straightforward (Udell et al., 2014).

Convexity also organizes structural graph optimization. A convex graph invariant is a graph invariant that is convex as a function of the adjacency matrix $$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$8. The elementary invariant is

$$cc_\lambda(cc_\mu(x,y),z)=cc_{\lambda\mu}\!\left(x,cc_{\tilde\mu}(y,z)\right),\qquad \tilde\mu=\frac{\lambda(1-\mu)}{1-\lambda\mu},$$9

and every convex graph invariant admits a representation

$\lambda\mu\neq 1$0

for suitable $\lambda\mu\neq 1$1 and scalars $\lambda\mu\neq 1$2 (Chandrasekaran et al., 2010). This yields invariant convex sets for maximum degree, spectral majorization, forbidden subgraph surrogates, and graph deconvolution; the same paper uses them in graph deconvolution, graph generation, and hypothesis testing between graph families (Chandrasekaran et al., 2010).

For projection problems, convexity connects set projection and multi-objective optimization. Given

$\lambda\mu\neq 1$3

the associated multi-objective convex problem is

$\lambda\mu\neq 1$4

The paper proves that exact solutions of the convex projection problem and the associated multi-objective problem coincide, and that approximate solutions transfer in both directions with sharp tolerance inflation factors $\lambda\mu\neq 1$5 and $\lambda\mu\neq 1$6 (Kováčová et al., 2021).

4. Convex geometries, representability, and neural codes

In closure theory, convexity appears as anti-exchange. A convex geometry is a closure system $\lambda\mu\neq 1$7 such that $\lambda\mu\neq 1$8 and, for every closed $\lambda\mu\neq 1$9 and distinct $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$0, the implication

$cc_\lambda(x,y)=\lambda x+(1-\lambda)y$1

holds (Adaricheva et al., 2022). For transit functions $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$2, the induced interval convexity $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$3 is the family of $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$4-convex sets $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$5 satisfying $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$6 for all $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$7. Under the Peano axiom $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$8, or more strongly under $cc_\lambda(x,y)=\lambda x+(1-\lambda)y$9, the paper on transit functions proves that

$X$0

(Changat et al., 2024). This unifies a range of graph convexities, including geodesic, monophonic, toll, weak toll, $X$1, $X$2, all-path, and cut-vertex convexities.

Representation theory separates small and large regimes. One paper proves that every finite convex geometry can be represented in the plane by a wide variety of convex sets extending Richter–Rogers’ polygon construction, but that general convex geometries cannot be represented by ellipses in the plane, and that there is no uniform bound on the number of common supporting lines allowed between pairs of representing convex sets; in higher dimensions every finite convex geometry of convex dimension $X$3 is representable in $X$4 by ellipsoids arbitrarily close to a ball (Kincses, 2017). On the other hand, for the special case of a 5-element base set, the paper on colors and ellipses proves that all $X$5 convex geometries admit a representation by ellipses, while several properties of circle geometries—the opposite property, nested triangle property, area $X$6 property, and separation property—obstruct circle representability; it also introduces colored-circle representations as unary predicates augmenting circle models (Adaricheva et al., 2022).

A simplicial-complex variant is convex union representability. A complex $X$7 is $X$8-convex union representable if it is the nerve of convex open sets in $X$9 whose union is itself convex. The paper “Convex Union Representability and Convex Codes” proves that not every collapsible complex has this property: there exist shellable collapsible complexes and non-evasive complexes that are not convex union representable (Jeffs et al., 2018). It also proves strong necessary conditions, including collapse onto the star of any face and collapsibility of the Alexander dual. For neural codes, the neural ideal

$$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$0

and its canonical form provide algebraic signatures of convexity and non-convexity. In particular, certain minimal pseudo-monomials in the canonical form detect disconnected restricted nerves or hollow simplices, and therefore non-convexity of the code (Curto et al., 2018).

5. Graph, network, and discrete-matrix convexity

In graph theory, a subgraph induced by a node set $$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$1 is convex if every geodesic path between any two nodes of $$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$2 lies entirely inside the induced subgraph. A connected network is called convex if every connected subset of nodes is convex (Marc et al., 2016). This notion yields several regimes. Trees and cliques are globally convex for opposite structural reasons; random graphs are only locally convex; and core-periphery networks can be regionally convex, with a non-convex core and convex periphery (Marc et al., 2016). The paper introduces convex-hull growth measures such as

$$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$3

and local-convexity scales

$$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$4

and reports that the Western US power grid, European highways, and a coauthorship graph are the most convex among the nine empirical networks studied, whereas the Little Rock food web is the only one classified as truly non-convex (Marc et al., 2016).

Directed graph convexities based on directed 2-paths lead to a sharper combinatorial theory. For an oriented graph $$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$5, the $$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$6-convexity forbids an outside vertex from being the center of a directed path $$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$7 with $$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$8; the $$\Delta_X=\left\{f:X\to[0,1]\mid f\text{ has finite support and }\sum_{x\in X}f(x)=1\right\},$$9-convexity imposes the same restriction only for induced directed paths, equivalently when $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$00 (Araújo et al., 23 Jun 2026). The paper proves that recognition of convex geometries is polynomial-time for $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$01-convexity, but coNP-complete for $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$02-convexity, even on DAGs. On the subclass of acyclic indifference oriented graphs, however, $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$03-geometricity is characterized by $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$04-freeness and becomes polynomial-time decidable (Araújo et al., 23 Jun 2026).

Discrete tomography yields yet another meaning. A $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$05-matrix is convex when the 1s are consecutive in every row and every column. Writing $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$06 for the convex matrices with row-sum vector $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$07 and column-sum vector $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$08, the paper on convex $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$09-matrices studies when such classes are nonempty and how individual matrices can be reconstructed (Brualdi et al., 2021). It extends ranked essential sets from permutation matrices to convex matrices, proves that the ranked essential set uniquely determines a matrix in $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$10, and gives an $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$11 reconstruction algorithm. It also shows, for example, that

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$12

iff $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$13 and

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$14

and uses the term epitope for information that uniquely determines a matrix in $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$15 (Brualdi et al., 2021).

6. Spacetime, imaging, and functional shape priors

In Lorentzian geometry, convexity must be adapted to indefinite signature. A smooth function $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$16 on a spacetime $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$17 is a spacetime convex function when its Hessian $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$18 has Lorentzian signature and satisfies

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$19

for all $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$20 (Gibbons et al., 2017). The level sets $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$21 then have second fundamental form controlled by

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$22

so spacetime convex functions generate foliations by expanding spacelike hypersurfaces (Gibbons et al., 2017). The paper proves that a spacetime admitting such a function has no closed spacelike geodesics, excludes certain closed marginally trapped surfaces, induces convex or subharmonic functions on special initial data sets, and exhibits barrier phenomena in the Schwarzschild interior, where $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$23 is a maximal hypersurface (Gibbons et al., 2017).

In data-driven image segmentation, convexity is imposed through quasi-concavity of the soft mask $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$24: all super-level sets

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$25

are required to be convex (Chen et al., 19 May 2026). This is equivalent to

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$26

The paper develops exact zero- and first-order characterizations and a second-order sufficient condition based on tangent-space negativity of the Hessian. In two dimensions, the practical sufficient quantity is

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$27

and the condition $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$28 on points where $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$29 implies quasi-concavity (Chen et al., 19 May 2026). This yields a differentiable loss

$cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$30

implemented by fixed finite-difference convolutions, together with a Convex Gradient Projection Module that performs an unrolled proximal refinement of the output mask. The paper reports that on Swin-Unet the second-order prior improves Dice by $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$31, IoU by $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$32, reduces Hausdorff distance by $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$33, and increases per-image runtime from $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$34 s to $cc_\lambda:C\times C\to C,\qquad \lambda\in[0,1],$35 s (Chen et al., 19 May 2026).

Across these settings, convexity is not a single definition but a stable structural pattern: closure under mixtures, interval containment, anti-exchange closure, geodesic preservation, Hessian positivity, or level-set quasi-concavity. The recent literature shows that these forms are tightly interconnected but not interchangeable: some are probabilistic, some possibilistic, some combinatorial, some Lorentzian, and some algorithmic. What remains common is that convexity converts local consistency conditions into strong global consequences—uniqueness, representability, tractable optimization, or topological rigidity (0903.5522, Changat et al., 2024, Gibbons et al., 2017).